Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 93 SPECIAL LIMITS 71EXAMPLESlirn x2 = +a and lim --x= +a.X+-OOx-+-UJIf a functionfdecreases without bound as its argument x increases without bound, we shall writelim f(x) = - CQ.X++COEXAMPLESlim -2x = -CO and lirn (1 -x2) = --COX'+QIf a function f decreases without bound as its argument x decreases without bound, we shall writelim f(x) = -W.x+-COEXAMPLESx++wEXAMPLE Consider the functionfsuch thatf(x) = x - [x] for all x. For each integer n, as x increases from n upto but not including n + 1, the value off@) increases from 0 up to but not including 1. Thus, the graph consists of asequence of line segments, as shown in Fig. 9-6. Then lirn f(x) is undefined, since the valuef(x) neither approachesx++ma fixed limit nor does it become larger or smaller without bound. Similarly, lim f(x) is undefined.x-+-OO-5 -4 -3 -2 -1 1 2 3 4 s xFig. 9-6Finding Limits at Infinity of Rational FunctionsA rational function is a quotient f(x)/g(x) of polynomials f(x) and g(x). For example,x2 - 5and are rational functions.4x' + 3x3x2 - 5x + 2x+7GENERALRULE. Tofind lirn fOand lirn - f(x), divide the numerator and the denominator byx-++oo g(x) x-'-Q) dx)the highest power of x in the denominator, and then use the fact thatfor any positive real number r and any constant c.CClim -=0 and lim -=0x++m x' x-t-a, x'